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Full text of "Treatise On Analysis Vol-Ii"

3    POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE               105

The notion of a positive measure enables us to define an order relation on
the vector space MR(X) of real measures on X. We write jj,^ v if the measure
v  ^ is positive. Since the relations /x^O and n^Q imply that /*(/) =
X/+) - X/~) =  for all/e ^n(X), and therefore that ^ = 0, it follows that
li < v is indeed an order relation on MR(X) (not, in general, a total ordering).
It is clear that \JL < v implies that 1 + v ^ A + v for all real measures A, and
0/x ^ (2v for all real scalars a * 0. (For a study of this order relation, see
(13.15).)

(13.3.2)   Let VL be a (complex) measure on X. Then there exists a smallest
positive measure p on X such that |//(/)| g p(\f\)for allfe Jf C(X).

For every positive measure v such that |//(/)| ^ v(|/|) for all/e
the relations g  0 and \h\ ^ g (g, h e JTC(X)) imply |ju(A)| g v(|A|)  v(#).
We shall show that there exists a positive measure p on X such that, for each
function/^ 0 in JTR(X), we have

(13.3.2.1)                       p(/)=        sup

This p will then clearly satisfy the conditions of (13.3.2).

To begin with, we remark that the right-hand side of (13.3.2.1) is finite;
for if K = Supp(/), then Supp(#) <= K, and

whenever \g\ ^/, by virtue of (13.1.2). Also it is clear that, for any real
scalar a g: 0, we have p(af) = ap(f). We shall show next that, if /t and /2
are two functions ^ 0 belonging to JTR(X), then

(13-3.2.2)                       p(/i+/2) = p(/i)

For each fi > 0, there exists g{ e ^TCW such
\9i\<,ft   and

Multiplying g{ by a complex number with absolute value 1, we may assume
that /<#,-) = \n(fft)\, and then we have

and since |^ +^2I Sa/i +/2 we have p(/t +/2) ^ p(/j) + p(/2) - 2e. Since
e is arbitrary, it follows that p(/i) + p(/2) ^ p(/i H-/2). On the other hand,
let h e Jf C(X) be such that \h\ ^/i -f /2 . Let ht be the function which is
equal to hfiKfi -f-/2) at the points x where /i(x) +/2(x) ^ 0 and is zero3.1.9) Let (Ua)a6l be an open covering ofX. For each a el let ua be a